MOMENTUM, HEAT AND MASS TRANSFER 317
If the shear stress at the surface isR 0 , the shearing force over areaAis the rate of
change of momentum or:
mus/tDR 0 A and Cp (^0) s/usDq/R 0
WritingR 0 DR, the shear stress acting on the walls andhas the heat transfer coef-
ficient between the fluid and the surface, then:
q/0sDhDR 0 Cp/usDRCp/us or h/CpusDR/u^2
From equation 3.19, the pressure change due to friction is given by:
PD 4 R/u^2 l/du^2
and substituting from equation 12.102:
PD 4 h/Cpul/du^2 D 4 hu/Cpl/d
The Reynolds analogy assumes no mixing with adjacent fluid and that turbulence
persists right up to the surface. Further it is assumed that thermal and kinematic equi-
libria are reached when an element of fluid comes into contact with a solid surface. No
allowance is made for variations in physical properties of the fluid with temperature.
A further discussion of the Reynolds analogy for heat transfer is presented in
Chapter 12.
Density of air at 320 KD 29 / 22. 4 273 ð 320 D 1 .105 kg/m^3.
The pressure drop:PD150 mm waterD 9. 8 ð 150 D1470 N/m^2
lD 4 .0m,dD 0 .050 m
In equation 3.23:
Pd^3 / 4 l^2 D 1470 ð 0. 0503 ð 1. 105 /[4ð 4. 0 0. 018 ð 10 ^3 ^2 ]
D 3. 192 ð 107
From Fig. 3.8, for a smooth pipe:ReD 1. 25 ð 105
and from Fig. 3.7:R/u^2 D 0. 0021
The heat transfer coefficient:
hDR/u^2 CpuD 0. 0021 ð 1. 05 ð 103 uD 2. 205 uW/m^2 K
Mass flowrate of air,GDu/ 4 0. 0502 D 0. 00196 ukg/s
Area for heat transfer,ADð 0. 050 ð 1. 0 D 0 .157 m^2.
GCpT 1 T 2 DhATmTw
whereT 1 andT 2 are the inlet and outlet temperatures andTmthe mean value taken as
arithmetic over the small length of 1 m.
∴ 0. 00196 uð 1. 050 ð 103 320 T 2 D 2. 205 uð 0. 157 0. 5 320 ðT 2 290
and: T 2 D316 K
The drop in temperature over the first metre is therefore 4 deg K.